Moiré phonons and impact of electronic symmetry breaking in twisted trilayer graphene

Twisted trilayer graphene is a particularly promising moir\'e superlattice system, due to its tunability, strong superconductivity, and complex electronic symmetry breaking. Motivated by these properties, we study lattice relaxation the long-wavelength phonon modes of this system. We show that mirror-symmetric hosts, aside from conventional acoustic modes, two classes shear which are even odd under mirror reflection. The mirror-even found be gapless equivalent "phason" twisted bilayer graphene, with appropriately rescaled parameters. have no analogue in exhibit finite gap, directly proportional degree relaxation. also discuss impact mirror-symmetry breaking, can tuned displacement field or result stacking shift, rotational- as well time-reversal-symmetry resulting spontaneous order. demonstrate induce angular momentum branches. Our findings important interpretation recent experiments, concerning origin superconductivity linear-in-$T$ resistivity.